- Use Euler's inequality (m≤3n-6) to prove that each planar graph has at least two vertices of degree at most 5. (Can you do better than 2?)

(b) Write down the degree of cach vertex in the following graph. Hence, verify the Handshaking Theorem.

Classify the graph given in each figure below as Eulerian, semi-Eulerian, or neither. Write the Eulerian circuit or Eulerian path if any exists. If your answer is neither Eulerian nor semi-Eulerian then give reason for your answer. (Letters indicate vertex names.)

Let G be a polyhedral graph with 12 vertices and 30 edges. Prove that the degree of each face of G is three. Show all of your work and clearly explain, using words, your reasoning.

Please, I want a correct solution to all paragraphs. The solution must be in text format and not handwritten Course name: computer drawing Question 3: A graph G has order 24 and size 72. The degree of each vertex is 3, 6 or 9. There are 6 vertices of degree six. How many vertices of degree 3 and 9?

4. Prove the following theorem: If a graph has more than two vertices of odd degree, then it does not have an Euler Path.   (Do a proof by contradiction please)

Prove that every simple, triangle-free, planar graph is 4-colorable. (Hint: show first that such a graph has a vertex of degree at most 3.)

Identify and write Euler circuit and Euler path. Also mention vertices, edges and degree of all vertices of the following graph. f A d D a B

Given a Triangle of three vertices A(2,3), B(4,2), C(3,0), Apply the shearing to the Triangle with SH=4

a. Show that K6.7 has a path containing all vertices in the graph. b. Explain why K6,7 is not Hamiltonian.